Update cheatsheet.fr.md

12.temporary_ins/04-math-tools/70.systems-of-differential-equations/40.parallel-1/cheatsheet.fr.md

@@ -70,46 +70,34 @@ $`M^0 = I_m`$ matrice identité de dimension $`m\times m`$.
 
 *$`\mathbf{M = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}}`$*
 
-**$`\mathbf{e^{\,M}}`$** $`\; = \sum_{n=0}^{+\infty}\dfrac{M^n}{n!}`$   
+**$`\mathbf{e^{\,M}}`$** $`\displaystyle\; = \sum_{n=0}^{+\infty}\dfrac{M^n}{n!}`$   
 <br>
-$`\begin{align} \quad\quad & = +\dfrac{1}{0!}\;\begin{pmatrix} 1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} + 
-+\dfrac{1}{1!}\;\begin{pmatrix}  \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix} + \cdots \\
+$`\begin{align} \quad\;\; & =\;\dfrac{1}{0!}\;\begin{pmatrix} 1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} + 
+\dfrac{1}{1!}\;\begin{pmatrix}  \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix} + \cdots \\
 & \\
-& \quad\quad \cdots +\dfrac{1}{2!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^2
+& \quad\;\; \cdots +\dfrac{1}{2!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^2
 +\cdots \\
 & \\
-& \quad\quad \cdots + \dfrac{1}{k!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^k + \cdots
+& \quad\;\; \cdots + \dfrac{1}{k!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^k + \cdots
 \end{align}`$    
 <br>
 <br>
-$`\begin{align} \quad\quad & = +\dfrac{1}{0!}\;\begin{pmatrix} 1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} + 
-+\dfrac{1}{1!}\;\begin{pmatrix}  \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix} + \cdots \\
+$`\begin{align} \quad\;\; & = \;\dfrac{1}{0!}\;\begin{pmatrix} 1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} + 
+\dfrac{1}{1!}\;\begin{pmatrix}  \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix} + \cdots \\
 & \\
-& \quad\quad \cdots +\dfrac{1}{2!}\;\begin{pmatrix} \lambda_1^2 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m^2 \\ \end{pmatrix}
+& \quad\;\; \cdots +\dfrac{1}{2!}\;\begin{pmatrix} \lambda_1^2 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m^2 \\ \end{pmatrix}
 +\cdots \\
 & \\
-& \quad\quad \cdots + \dfrac{1}{k!}\;\begin{pmatrix} \lambda_1^k & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m^k \\ \end{pmatrix} + \cdots
+& \quad\\;\; \cdots + \dfrac{1}{k!}\;\begin{pmatrix} \lambda_1^k & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m^k \\ \end{pmatrix} + \cdots
 \end{align}`$    
 <br>
+<br>
+$`\quad\;\; & = \;\begin{pmatrix} \dfrac{1}{n!}\sum_{n=0}^{+\infty}\lamnda_1^n & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \dfrac{1}{n!}\sum_{n=0}^{+\infty}\lamnda_m^n \\ \end{pmatrix}`$
 
 
 
 
 
-$`\begin{align}\quad\quad\quad & = \sum_{n=0}^{+\infty}\dfrac{M^n}{n!} \\
-& \\
-& = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} + 
-\begin{pmatrix}  \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix} + \cdots \\
-& \\
-& \quad\quad \cdots +\dfrac{1}{2!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^2
-+\cdots \\
-& \\
-& \quad\quad \cdots + \dfrac{1}{k!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^k + \cdots
-\end{align}`$
-
-
-
-